BUS7010 Quant Prep Statistics in Business and Economics Week 5 Dr. Jenne Meyer BUS7010 Quant Prep Statistics in Business and Economics

Measure of Central Tendency A single value that summarizes a set of data. It locates the center of the values Arithmetic mean Weighted mean Median Mode Geometric mean

ARITHMETIC MEAN ARITHMETIC MEAN Pop mean = sum of all the values in pop # of values in the pop µ = ∑X N

Properties of arithmetic mean Every set of interval data has a mean All values are included Mean is unique - only one Useful to compare two or more populations Sum of the deviations of each value from the mean will always be zero Disadvantage of arithmetic mean Mean may not be representative Can’t use for open-ended (range) data

Median The midpoint of the values (exactly half are below, half are above) Used when the mean is not representative due to high value outliers Unique number Not affected by extremely large or small values Can be used with open-ended range values Can be used for several measurement types

Mode The value that appears most frequently Can be used fir any measurement type Not affected by extremely large or small values Sometimes it doesn’t exist Sometimes it represents more than one value

Formulas in Excel

Skewness – Mean, Median, Mode

Measures of Dispersion Range Mean deviation Variance Standard deviation Range = highest value – lowest value Mean deviation – the arithmetic mean of the absolute values of the deviations from the mean The # deviates of average x amount from the mean Variance – the arithmetic mean of the squared deviations from the mean Compare the dispersion of two or more sets of data Standard deviation – the square root of the variance represents the spread or variability of the data, the average range from the center point

Variation Population variation =varp(…) Sample variation =var(…)

Standard Deviation Population variation Sample variation =stdevp(…)

Sample Standard Deviation Sample standard deviation is most common use of statistics

Standard Deviation Example: Numbers Mean Standard Deviation 100,100,100,100,100,100 100 0 90, 90, 100, 110, 110 100 10 Computing the standard deviation: find the mean (100) find the deviation/variance of each value form the mean (-10, -10, 0, 10, 10) square the deviations/variances (100, 100, 0, 100, 100) sum the squared deviations (100+100+0+100+100 = 400) divide the sum by the # of values minus 1 (# of values = 5 – 1 = 4, 400/4 = 100) take the square root of the variance (10) (Will be important in research when you are trying to determine the range of information.)

Coefficient of Variation To compare dispersion in data sets with dissimilar units of measurement (e.g., kilograms and ounces) or dissimilar means (e.g., home prices in two different cities) we define the coefficient of variation (CV), which is a unit-free measure of dispersion:

Formulas in Excel

Time Series Analysis

Frequency curves Normal distribution

Central Limit Theorem Chebyshev’s Theorem If all samples of a particular size are selected from any population, the sampling distribution of the sample mean is approximately a normal distribution. This approximation improves with larger samples. (the larger the sample, the more it appears to be a normal standard distribution)

Central Limit Theorem Chebyshev’s Theorem

Central Limit Theorem Chebyshev’s Theorem

Standard Normal Distribution Z value – converts the actual distribution to a standard distribution. (It is the distance between the selected value (x) and the mean (µ) divided by the standard deviation (σ) Normal distributions can be transformed to standard normal distributions by the formula: A “Z” score always reflects the number of standard deviations above or below the mean a particular score is A person scored 60 on a test with a μ=50 and σ=10, then he scored 1 standard deviations above the mean. Converting the test score to a Z score, an X of 70 would be: Z=1=0.3413 11/28/2018

Standard Normal Distribution Standard Normal Table (once z is computed) A table of probabilities for a Z random variable. See page 479 11/28/2018

Example p 224/5, likelihood of finding a foreman w/ a salary between $1000 and $1100 is 34.13%

Standard Normal Distribution p227 11/28/2018